Homework 2 Cover Sheet
Name:
Space below is for the instructor.
Problem
Score
CDS 140a: Homework Set 2
Due: Wednesday, October 17th, 2007.
Dierential Equations and Dynamical Systems, Lawrence Perko, Third Edition.
1.3: 1(c) without using the hint.
1.6:

Homework 4 Cover Sheet
Name:
Space below is for the instructor.
Problem
Score
CDS 140a: Homework Set 4
Due: Wednesday, October 31, 2007.
Nonlinear Dierential Equations and Dynamical Systems, Ferdinand Verhulst, Second Edition.
Chapter 3: 1, 3, 4, 5
Chap

Homework 6 Cover Sheet
Name:
Space below is for the instructor.
Problem
Score
CDS 140a: Homework Set 6
Due: Wednesday, November 21, 2007.
Nonlinear Dierential Equations and Dynamical Systems, Ferdinand Verhulst, Second Edition.
Consider two systems x = A

Homework 7 Cover Sheet
Name:
Space below is for the instructor.
Problem
Score
CDS 140a: Homework Set 7
Due: Wednesday, November 28, 2007.
Describe the bifurcation at the origin for the following system
x = 2x + 3y + x + y 3
y = 2x 3y + x3
Describe the b

Page 67
Invariant Manifolds
There are two basic motivations for invariant manifolds. The rst comes
from the notion of separatrices that we have seen in our study of planar
systems, as in the gures. We can ask what is the higher dimensional generalization

Page 1
Invariant Manifolds and Liapunov
Functions
Invariant Manifolds
The motivation for invariant manifolds comes from the study of critical
elements of linear dierential equations of the form
x = Ax,
x Rn .
Let E s , E c , and E u be the (generalized) r

Introduction
82
Liapunov Functions
Besides the Liapunov spectral theorem, there is another basic method of
proving stability that is a generalization of the energy method we have seen
in the introductory mechanical examples.
Denition (Liapunov Function).

Linear Systems Notes for CDS-140a
October 27, 2008
1
Linear Systems
Before beginning our study of linear dynamical systems, it only seems fair to ask
the question why study linear systems? One might hope that most/all realworld systems are linear systems,

1.6 Mechanical Systems
1.6
49
Mechanical Systems
Mechanics provides an important and large class of systems for both motivating the ideas of dynamical systems and to which the ideas of dynamical
systems apply.
We saw some simple examples of Newtonian mech

1.6 Mechanical Systems
1.6
45
Mechanical Systems
Mechanics provides an excellent class of systems for both motivating the
ideas of dynamical systems and to which the ideas of dynamical systems
apply.
We saw some simple examples of Newtonian mechanical sys

CDS 140a Midterm Examination Policy
1. The exam is due by 5pm Tuesday, November 6, 2007 in Steele 3.
2. You shall abide by the Caltech Honor Code1 which states
No member of the Caltech community shall take unfair advantage of any
other member of the Calte

Normal form for a 2D diagonalizable system
Sujit Nair
Consider the following example.
x
y
=
1 0
0 2
x
y
+
f (x, y )
g(x, y )
(1)
Consider the following change of coordinates.
x
y
=
u
v
+ h(u, v )
(2)
The operator LJ (h(u, v ) for this example is
LJ (h(u,

42
1.4
Introduction
Stability and Linearization
Suppose we are studying a physical system whose states x are described
by points in Rn . Assume that the dynamics of the system is governed by a
given evolution equation
dx
= f (x).
dt
Let x0 be a stationary

1.4 Stability and Linearization
1.4
39
Stability and Linearization
Suppose we are studying a physical system whose states x are described
by points in Rn . Assume that the dynamics of the system is governed by a
given evolution equation
dx
= f (x).
dt
Let

A LITTLE BIT ON SYMMETRIES
HENRY JACOBS
Consider a dynamical system on Rn
(1)
x = f (x)
Let x(t) be a trajectory that satises the ODE. Consider a transformation, , from
trajectories to trajectories. That is to say, if is the set of curves on Rn , then :

1.3 Vector Fields and Flows.
1.3
21
Vector Fields and Flows.
This section introduces vector elds on Euclidean space and the ows they
determine. This topic puts together and globalizes two basic ideas learned
in undergraduate mathematics: the study of vect

Homework 3 Cover Sheet
Name:
Space below is for the instructor.
Problem
Score
CDS 140a: Homework Set 3
Due: Wednesday, October 24th, 2007.
Nonlinear Dierential Equations and Dynamical Systems, Ferdinand Verhulst, Second Edition.
Chapter 2: 1, 2, 3, 4, 5,

CDS140a - Introduction to Dynamics
Homework 1
Exercises 1, 2, and 5
Fernando Ferrari de Goes
October 7, 2009
1. Consider the following planar system for (x, v ) R2 :
x=v
v = x3
(1)
(a) Find the equilibrium points for the system.
The equilibrium points of

1
CDS 140a: Homework Set 2
Due: Friday, October 16th, 2009. 1. Verify directly that the homoclinic orbits for the simple pendulum equation + sin = 0 are given by (t) = 2 tan1 (sinh t). 2. Derive the same formula as in Exercise 1 using conservation of ener

1
CDS 140a: Homework Set 3
Due: Friday, October 23th, 2009. 1. Solve the system x=xy y = x + 3y for given initial conditions (x0 , y0 ). 2. Do all solutions of the system x = x + y + z y = y + 2z z = 2z converge to the origin as t ? 3. Do all solutions of

1
CDS 140a: Homework Set 4
Due: Friday, October 30st, 2009. 1. Show that if A is diagonalizable, then det eA = etrace A Try this out on a few nondiagonalizble matrices and make a conjecture as to its general validity. 2. Solve the system x = ax by y = bx

1
CDS 140a: Homework Set 5
Due: Friday, November 6, 2009. 1. (Computer work). Plot the phase portrait of the van der Pol oscillator x=v v = x + v (1 x2 ) and conjecture about its global structure. Do you think that solutions exist for all time? 2. Lineari

1
CDS 140a: Homework Set 6
Due: Friday, November 20, 2009. 1. Let a be a real parameter with 0 a 4. The logistic map is the map of the unit interval [0, 1] to itself that is dened by f (x) = ax(1 x). Find the xed points of f and determine their stability.

1
CDS 140a Final Examination J. Marsden, December, 2008
Attempt SEVEN of the following ten questions. Each question is worth 20 points. The exam time limit is three hours; no aids are permitted. Turn in the exam by 5pm on Thursday, December 11, 2008.
Prin

1
CALIFORNIA INSTITUTE OF TECHNOLOGY
Control and Dynamical Systems
CDS 140a Midterm Examination
Jerry Marsden
Nov. 5, 2009
This is a three hour, closed book exam
While no aids are permitted, results from the course may be used
as long as they are accurate

1
CDS 140a Final Examination
J. Marsden, December, 2009
The Exam is due by noon on Friday, December 11, 2009
Attempt SEVEN of the following ten questions.
Each question is worth 20 points.
The exam time limit is three hours; no aids of any kind
(including

Page 1
1
Basic Theory of Dynamical Systems
1.1
Introduction and Basic Examples
Dynamical systems is concerned with both quantitative and qualitative
properties of evolution equations, which are often ordinary dierential equations and partial dierential eq

1
CALIFORNIA INSTITUTE OF TECHNOLOGY
Control and Dynamical Systems
CDS 140a Midterm Examination
Jerry Marsden
Nov. 6, 2008
This is a three hour, closed book exam
While no aids permitted, results from the course may be used, but
they must be quoted.
Turn i